This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A059365 #47 Sep 08 2022 08:45:02
%S A059365 0,0,1,0,1,1,0,2,2,1,0,5,5,3,1,0,14,14,9,4,1,0,42,42,28,14,5,1,0,132,
%T A059365 132,90,48,20,6,1,0,429,429,297,165,75,27,7,1,0,1430,1430,1001,572,
%U A059365 275,110,35,8,1,0,4862,4862,3432,2002,1001,429,154,44
%N A059365 Another version of the Catalan triangle: T(r,s) = binomial(2*r-s-1,r-1) - binomial(2*r-s-1,r), r>=0, 0 <= s <= r.
%H A059365 G. C. Greubel, <a href="/A059365/b059365.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059365 F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.
%H A059365 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive bijective approach to counting permutations containing 3-letter patterns</a>, arXiv:math/0211380 [math.CO], 2002.
%H A059365 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000011">The number of touch points of a Dyck path.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000025">The number of initial rises of a Dyck paths.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000061">The number of nodes on the left branch of the tree.</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000084">The number of subtrees.</a>
%H A059365 A. Robertson, D. Saracino and D. Zeilberger, <a href="https://arxiv.org/abs/math/0203033">Refined restricted permutations</a>, arXiv:math/0203033 [math.CO], 2002.
%H A059365 <a href="/index/Ca#Catalan">Index to sequences related to Catalan</a>
%F A059365 Essentially the same triangle as [0, 1, 1, 1, 1, 1, 1, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938, but the first term is T(0,0) = 0.
%e A059365 Triangle starts
%e A059365 0;
%e A059365 0, 1;
%e A059365 0, 1, 1;
%e A059365 0, 2, 2, 1;
%e A059365 0, 5, 5, 3, 1;
%e A059365 0, 14, 14, 9, 4, 1;
%e A059365 0, 42, 42, 28, 14, 5, 1;
%e A059365 0, 132, 132, 90, 48, 20, 6, 1;
%e A059365 0, 429, 429, 297, 165, 75, 27, 7, 1;
%e A059365 0, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1;
%e A059365 0, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1;
%e A059365 ...
%t A059365 Table[Binomial[2*r - s - 1, r - 1] - Binomial[2*r - s - 1, r], {r, 0, 10}, {s, 0, r}] // Flatten (* _G. C. Greubel_, Jan 08 2017 *)
%o A059365 (PARI) tabl(nn) = { print(0); for (r=1, nn, for (s=0, r, print1(binomial(2*r-s-1,r-1)-binomial(2*r-s-1,r), ", ");); print(););} \\ _Michel Marcus_, Nov 01 2013
%o A059365 (Magma) /* as triangle */ [[[0] cat [Binomial(2*r-s-1, r-1)- Binomial(2*r-s-1, r): s in [1..r]]: r in [0..10]]]; // _Vincenzo Librandi_, Jan 09 2017
%Y A059365 See also the triangle in A009766. First 2 diagonals both give A000108, next give A000245, A002057.
%Y A059365 Cf. A009766, A000007, A084938, A000108.
%Y A059365 The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
%Y A059365 Essentially the same as A033184.
%Y A059365 The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A053121, A059365, A099039, A106566, A130020, A047072, A171567, A181645.
%Y A059365 Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
%K A059365 nonn,tabl
%O A059365 0,8
%A A059365 _N. J. A. Sloane_, Jan 28 2001